Sparse Signal Reconstruction Based on Multiparameter Approximation Function with Smoothed 0 Norm

Abstract

The smoothed 0 norm algorithm is a reconstruction algorithm in compressive sensing based on approximate smoothed 0 norm. It introduces a sequence of smoothed functions to approximate the 0 norm and approaches the solution using the specific iteration process with the steepest method. In order to choose an appropriate sequence of smoothed function and solve the optimization problem effectively, we employ approximate hyperbolic tangent multiparameter function as the approximation to the big “steep nature” in 0 norm. Simultaneously, we propose an algorithm based on minimizing a reweighted approximate 0 norm in the null space of the measurement matrix. The unconstrained optimization involved is performed by using a modified quasi-Newton algorithm. The numerical simulation results show that the proposed algorithms yield improved signal reconstruction quality and performance.

1. Introduction

Sparse representation of signals has been extensively studied for decades. The concept of signal sparsity and 1 norm based on recovery techniques can be traced back to the work of Logan [1] in 1965, Santosa and Symes [2] in 1986, and Donoho and Stark in 1989 [3]. It is generally agreed that the foundation of the current state of the art in compressive sensing (CS) theory was laid by Candés et al. [4], Donoho [5], and Candés and Tao [6] in 2006. Mathematically, under sparsity assumptions, one would want to recover a signal 𝑥∈ℝ^𝑛, for example, the coefficient sequence of the signal in the appropriate basis, by solving the combinatorial optimization problem: min_𝑥‖𝑥‖₀s.t.𝑦=Ψ𝑥, (1) where 𝑦 is an observed signal, Ψ is the dictionary matrix, 𝑥 means a coefficient vector for the linear combination, and ‖𝑥‖₀ is the 0 norm of the vector 𝑥 indicating the number of its nonzero elements.

Exactly solving (1) is reported to be NP-hard [5, 7]; thus a relaxed approach [8–10] was proposed to address this challenge. By replacing nonconvex 0 norm with convex 1 norm, the method finds optimal solution using convex optimization [8, 11–13]. It was also introduced [6, 14–17] that, for many problems, the minimizing of the 1 norm is equivalent to that of the 0 norm under certain conditions. However, experimental results in [13] posed a strong question about the equivalence of minimizing both norms in practical problems. The sparse signal reconstruction (SSR) algorithm based on the optimization of a smoothed approximate 0 norm is studied in [11] where simulation results are compared with corresponding results obtained with several existing SSR algorithms with respect to reconstruction performance and computational complexity. These results favor the use of the approximate 0 norm.

In this paper, we present a new signal reconstruction algorithm for CS, which is based on the minimization of a smoothed approximate 0 norm. But it differs from the previous algorithms in several aspects. First, we use approximation hyperbolic tangent function to approximate the 0 norm, which is found to have better approximation quality of 0 norm than the standard Gauss function. Second, the 0 norm minimization algorithm in this paper is carried out in null space of the measurement matrix, in which the constraints on measurements are eliminated and become unconstrained. Third, we use a reweighting technique in the null space of the measurement matrix to force the algorithm to reach the desired sparse solution faster. The rest of the paper is organized as follows. In Section 2, the smoothed 0 norm algorithm is briefly presented. Section 3 is about the signal reconstruction on the base of the reweighted approximation smoothed 0 norm algorithm. Section 4 provides the experimental results. The paper is finally concluded in Section 5.

2. Smoothed 0 Norm Algorithm (SL0)

Smoothed 0 norm attempts to solve the problem in (1) by approximating the 0 norm with a continuous function. Consider the continuous Gaussian function 𝑓_𝜎(𝑥) with the parameter 𝜎: 𝑓_𝜎(𝑥)=𝑒^{−𝑥2/2𝜎2},𝑥∈ℝ,𝜎∈ℝ₊. (2)

The parameter 𝜎 determines the quality of the approximation and for any family of functions 𝑓_𝜎(𝑥) which approximates the Kronecker delta function.

Define the continuous multivariate function 𝑔(𝑥) as 𝑔(𝑥)=^𝑁_𝑖=1𝑓_𝜎𝑥_𝑖,𝑥∈ℝ^𝑁×1. (3) It follows from (2) and (3) that 𝜎→0 and lim_𝜎→0𝑓_𝜎(𝑥)=1𝑥=00𝑥≠0. (4)

Since the number of entries in 𝑥 is 𝑁 and the function 𝑔(𝑥) is an indicator of the number of zero entries in 𝑥, the 0 norm of reconstructed vector 𝑥 is approximated by ‖𝑥‖₀≈𝑁−𝑔(𝑥). (5)

Substituting this approximation into (1) yields the problem minimise^𝑁_𝑖=11−𝑒^{−𝑥2𝑖/2𝜎2}s.t.𝑦=Ψ𝑥. (6)

The approach is then to solve the problem (6) for a decreasing sequence of 𝜎’s. The underlying thought is to select a 𝜎 which ensures that the initial solution is in the subset of ℝ^𝑁×1 over which the approximation is convex and gradually increases the accuracy of the approximation. Here, the final smoothed 0 algorithm is proved to be faster with the possibility of recovering a sparser solution [11].

Algorithm 1. The final smoothed 0 norm (SL0 algorithm) is as follows.(1)Initialization: choose an arbitrary solution from the feasible set, for example, the minimum 2 norm solution of 𝑥=Ψ𝑥; that is, 𝑥=Ψ^𝑇ΨΨ^𝑇−1𝑦. (7)(2)Choose a decreasing sequence for 𝜎_𝑖, 𝑖=1,2,…,𝐾.(3)For 𝑘=1,2,…,𝐾, let 𝜎=𝜎_𝑘. Minimize function (6) using 𝐿 iterations of the steepest ascent algorithm.(a)Let Δ𝑥=[𝑥₁exp(−𝑥²₁/2𝜎²_𝑘),…,𝑥_𝑛exp(−𝑥²_𝑛/2𝜎²_𝑘)]^𝑇.(b)Let 𝑥←𝑥−𝜇Δ𝑥 (where 𝜇 is a small position constant); then 𝑥⟵𝑥−Ψ^𝑇ΨΨ^𝑇−1𝑦. (8)(4)Final answer is 𝑥.

3. Signal Reconstruction Based on Approximation Function

3.1. Approximate 0 Norm with Approximate Hyperbolic Tangent Sequence

The key to SL0 algorithm is to select high quality smoothed continuous function to approximate 0 norm. The optimal value of 0 norm is achieved through the solution of the minimum value of the approximate smoothed continuous function. In general, the approximation of the 0 norm with Gauss function cannot produce desired result. To effectively approximate 0 norm, we analyzed inverse tangent function [18], hyperbolic tangent function, and approximate hyperbolic tangent sequence.

Inverse tangent function is as follows: ℎ_𝜎𝑥_𝑖=2𝜋𝑥arctan^2𝑖2𝜎². (9) Hyperbolic tangent function is as follows: 𝑔_𝜎𝑥_𝑖=𝑒^{𝑥2𝑖/2𝜎2}−𝑒^{−𝑥2𝑖/2𝜎2}𝑒^{𝑥2𝑖/2𝜎2}+𝑒^{−𝑥2𝑖/2𝜎2}. (10) Approximate hyperbolic tangent function is as follows: 𝑓_𝜎𝑥_𝑖=18𝑒^{𝑥2𝑖/2𝜎2}−𝑒^{−𝑥2𝑖/2𝜎2}𝑒^{𝑥2𝑖/2𝜎2}+𝑒^{−𝑥2𝑖/2𝜎2}+781−𝑒^{−𝑥2𝑖/((1/8)𝜎2)}, (11) where 𝜎 is a parameter and 𝑥_𝑖 is a component of the vector 𝑥. The following properties can be obtained.

(1) For formula (11), let 𝐹_𝜎(𝑥)=∑^𝑁_𝑖=1𝑓_𝜎(𝑥_𝑖); then ‖𝑥‖₀=lim_𝜎→0𝐹_𝜎(𝑥) since lim_𝜎→0𝑓_𝜎𝑥_𝑖=lim_𝜎→018⋅𝑒^{𝑥2𝑖/2𝜎2}−𝑒^{−𝑥2𝑖/2𝜎2}𝑒^{𝑥2𝑖/2𝜎2}+𝑒^{−𝑥2𝑖/2𝜎2}+78⋅1−𝑒^{−𝑥2𝑖/((1/8)𝜎2)}=0𝑥_𝑖=01𝑥_𝑖≠0. (12) According to the definition of 0 norm, when 𝜎 goes infinitely to zero, the value of 𝐹_𝜎(𝑥) goes as well to the nonzero elements of the vector 𝑥, that is, the 0 norm of vector 𝑥. Therefore, it can be shown with the smoothed function ‖𝑥‖₀=lim_𝜎→0𝐹_𝜎(𝑥). Obviously, formulas (9) and (10) have the same approximation property of 0 norm as that of (11). The approximation performance of 0 norm of each formula, however, is different. For further illustration of the advantages of approximate hyperbolic tangent function as smoothed continuous function, we compared the distribution of each approximation function at interval [−1,1] when 𝜎=0.1. Figure 1 is the approximation quality comparison of the smoothed 0 norm of the standard Gauss function, inverse tangent function, hyperbolic tangent function, and approximate hyperbolic tangent function. The estimation of 0 norm by approximate hyperbolic tangent function outsmarted other functions. It shows that the approximate hyperbolic tangent function has better “steep nature" between the intervals of [−0.5,0.5] and the estimation of the 0 norm is more accurate.

Figure 1 

Comparison of the four approximate ℓ0 norm functions with 𝜎=0.1.

(2) Given 𝜎≥0, approximate hyperbolic tangent function 𝑓_𝜎(𝑥_𝑖) has better convergence than other approximate functions; that is, 𝑓_𝜎(𝑥_𝑖)≥(1−𝜑(𝑥_𝑖)), for arbitrary 𝜎, 𝜑, where 𝜑(𝑥_𝑖)=𝑒^{−𝑥2𝑖/2𝜎2}.

Proof. Let 𝑢(𝑥)=𝑓(𝑥)−(1−𝜑(𝑥)), when 𝑥=0 or 𝑥=∞, and 𝑢(𝑥)=0, when 0<|𝑥|<∞, 𝑢(𝑥)=18⋅𝑒^𝑥2/2𝜎2−𝑒^{−𝑥2/2𝜎2}𝑒^𝑥2/2𝜎2+𝑒^{−𝑥2/2𝜎2}+78⋅1−𝑒^{−𝑥2/((1/8)𝜎2)}−1+𝑒^{−𝑥2/2𝜎2}=18⋅1−2𝑒^{−𝑥2/2𝜎2}𝑒^𝑥2/2𝜎2+𝑒^{−𝑥2/2𝜎2}+78⋅1−𝑒^{−𝑥2/((1/8)𝜎2)}−1+𝑒^{−𝑥2/2𝜎2}1=−8⋅2𝑒^{−𝑥2/2𝜎2}𝑒^𝑥2/2𝜎2+𝑒^{−𝑥2/2𝜎2}−78⋅𝑒^{−𝑥2/((1/8)𝜎2)}+18⋅𝑒^{−𝑥2/2𝜎2}+78⋅𝑒^{−𝑥2/2𝜎2}=18⋅−2𝑒^{−𝑥2/2𝜎2}𝑒^𝑥2/2𝜎2+𝑒^{−𝑥2/2𝜎2}+𝑒^{−𝑥2/2𝜎2}+78⋅𝑒^{−𝑥2/2𝜎2}−𝑒^{−𝑥2/((1/8)𝜎2)}=18⋅𝑒^{−𝑥2/2𝜎2}𝑒^𝑥2/2𝜎2+𝑒^{−𝑥2/2𝜎2}⋅−2+𝑒^𝑥2/2𝜎2+𝑒^{−𝑥2/2𝜎2}+78⋅𝑒^{−𝑥2/2𝜎2}−𝑒^{−𝑥2/((1/8)𝜎2)}≥18⋅11+𝑒^𝑥2/𝜎2𝑒^𝑥2/4𝜎2+𝑒^{−𝑥2/4𝜎22}≥0. (13)

From (9), (10), and (11), it is clear that the smaller the 𝜎 is, the more the local extreme values of objective function would be, and it would be difficult to get the global optimal value of the objective function. The 𝜎 determines the smoothness of the objective function. The bigger the 𝜎, the smoother the objective function, and the less accurate it would be to estimate the 0 norm. Conversely, the smaller the 𝜎 is, the better the approximation of the 0 norm would be. In the process, we built a decreasing sequence 𝜎₁,𝜎₂,…,𝜎_𝑛 to optimize each corresponding objective function until 𝜎_𝑖 is small enough, so as to eliminate the influence of the local extreme value and get the global optimal value of the smoothed function. We, thus, use approximate hyperbolic tangent sequence for the approximation of 0 norm.

Problem (1) can be approximated by the following model: min𝐹_𝜎(𝑥)s.t.𝑦=Φ𝑥. (14)

(3) There are lots of algorithms to the solving of problem (14), among which the most representative one is the steepest descent method. But there still exist two drawbacks.(a)“Notched effect" in search direction strongly hinders the convergence speed.(b)Search steps cannot be estimated. Usually, it is estimated with experiences, which lacks theoretical support.

We, therefore, propose the modified quasi-Newton algorithm to solve the above problem. First, calculate the Newton direction of the approximate hyperbolic tangent function: 𝑑=−∇²𝐹_𝜎(𝑥)⁻¹∇𝐹_𝜎(𝑥), (15) where ∇𝐹_𝜎(𝑥)=18⋅4𝑥₁/𝜎²𝑒^{𝑥21/2𝜎2}+𝑒^{−𝑥21/2𝜎22}+78⋅16𝑥₁𝜎²𝑒^{−8𝑥21/𝜎2}1,…,8⋅4𝑥_𝑁/𝜎²𝑒^{𝑥2𝑁/2𝜎2}+𝑒^{−𝑥2𝑁/2𝜎22}+78⋅16𝑥_𝑁𝜎²𝑒^{−8𝑥2𝑁/𝜎2𝑇},∇²𝐹_𝜎(𝑥)=𝜕²𝑓_𝜎𝑥₁𝜕𝑥²₁0𝜕0⋯0₂𝑓_𝜎𝑥₂𝜕𝑥²₂𝜕⋯0⋮⋮⋮⋮00⋯²𝑓_𝜎𝑥_𝑁𝜕𝑥²_𝑁,𝜕²𝑓_𝜎𝑥_𝑘𝜕𝑥²_𝑘=18⋅4/𝜎²1+𝑥²_𝑘/𝜎²𝑒^{𝑥2𝑘/2𝜎2}+𝑒^{−𝑥2𝑘/2𝜎22}+78⋅16𝜎²𝜎^{−8𝑥2𝑘/𝜎2}1−16𝑥²_𝑘𝜎². (16)(17)(18) We know that the Hesse matrix is nonpositive definite in Newton direction 𝑑, which cannot ensure that the Newton direction is descent direction, so the Hesse matrix should be modified. Then, set up a new matrix: 𝐺=∇²𝐹_𝜎(𝑥)+𝜀_𝑘𝐼, (19) where 𝜀_𝑘 is a group of proper positive numbers, 𝐼 is identity matrix, and the diagonal elements in matrix 𝐺 are all positive. Since lim_𝑥𝑘→0𝑒^{𝑥2𝑘/2𝜎2}+𝑒^{−𝑥2𝑘/2𝜎22}=4, (20) choose 𝜀_𝑘=78⋅16²𝑥²_𝑘𝜎⁴𝑒^{−8𝑥2𝑘/𝜎2}, (21) as modification coefficients; then the diagonal elements in matrix 𝐺 can be shown as 18⋅41+𝑥²_𝑘/𝜎²𝜎²𝑒^{𝑥2𝑘/2𝜎2}+𝑒^{−𝑥2𝑘/2𝜎22}+78⋅16𝜎²𝑒^{−8𝑥2𝑘/𝜎2}. (22) Also, it can be obtained that 𝑑=−∇²𝐹_𝜎(𝑥)⁻¹⋅∇𝐹_𝜎(𝑥)=−𝜎²𝑥₁+28⋅4𝜎²𝑥₁𝑒^{−8𝑥21/𝜎2}𝜎²+𝑥²₁+28⋅4𝜎²𝑒^{−8𝑥21/𝜎2}−𝜎,…,₂𝑥_𝑁+28⋅4𝜎²𝑥_𝑁𝑒^{−8𝑥2𝑁/𝜎2}𝜎²+𝑥²_𝑁+28⋅4𝜎²𝑒^{−8𝑥2𝑁/𝜎2𝑇}. (23)

(4) Usually, parameter 𝜎 is chosen as 𝜎_𝑗=𝛾⋅𝜎_𝑗−1, 𝑗=1,2,…, and 𝐽,𝛾∈(0.5,1). 𝜎₁ and 𝜎_𝑗 are chosen as follows.

Let 𝑥=max_𝑖|𝑥⁰_𝑖|, for faster algorithm convergence; parameter 𝜎 should meet the following: ℎ_𝜎1(𝑥)=2𝜋𝑥arctan²2𝜎²₁≤12⟹𝜎₁≥𝑥√2, (24) and choosing 𝜎₁=max_𝑖(|𝑥⁰_𝑖|/√2), when 𝜎_𝑗→0, we can obtain 𝐹_𝜎𝑗(𝑥)→‖𝑥‖₀, so when 𝜎_𝑗 is smaller, 𝐹_𝜎𝑗(𝑥) can better reflect the sparsity of vector 𝑥. Meanwhile, however, it is more sensitive to noise; the value of 𝜎_𝑗 should not be too small. In summary, the steps are as follows.

Algorithm 2. The approximate smoothed 0 norm (ASL0 algorithm) is as follows. (1)Initialization: 𝑥=Φ^𝑇𝑦, initial residual 𝑟₀=0, and smoothed function’s initial parameter 𝜎=1.(2)Calculate the modified Newton direction with (23) and use modified Newton algorithm to get 𝑥=𝑥+𝑑.(3)Use gradient projection to have 𝑥=𝑥−Φ^𝑇(ΦΦ^𝑇)⁻¹(Φ𝑥−𝑦), and calculate the residual 𝑟=𝑦−Φ𝑥.(4)If the residual of the adjacent iterations meets ‖𝑟−𝑟₀‖≤𝜀, then stop. Otherwise, let 𝑟₀=𝑟, continue, and return to (2).(5)Further decrease 𝜎, let 𝜎=𝛾⋅𝜎(0<𝛾<1), usually set 𝛾=0.7, and return to (2). Otherwise, stop iterating and get the optimal value 𝑥.

3.2. Reweighting the Approximate 0 Norm Minimization

It is well known that all solutions of Ψ𝑥=𝑦 can be parameterized as 𝑥=𝑥_𝑠+𝑉_𝑛𝜉, (25) where 𝑥_𝑠 is a solution of Ψ𝑥=𝑦 and 𝑉_𝑛 is a matrix whose columns constitute an orthogonal basis of the null space of Ψ [19]. Vector 𝑥_𝑠 and matrix 𝑉_𝑛 can be evaluated by using the singular-value decomposition (SVD) or by using the QR decomposition of matrix Ψ [20]. Using (25) the problem in (14) is reduced to minimize_𝜉^𝑁_𝑖=1𝑓_𝜎𝑥_𝑠𝑖+𝑉^𝑇_𝑖𝜉, (26) where 𝑓_𝜎𝑥_𝑠𝑖+𝑉^𝑇_𝑖𝜉=18⋅𝑒^{[𝑥𝑠𝑖+𝑣𝑇𝑖𝜉]2/2𝜎2}−𝑒^{−[𝑥𝑠𝑖+𝑣𝑇𝑖𝜉]2/2𝜎2}𝑒^{[𝑥𝑠𝑖+𝑣𝑇𝑖𝜉]2/2𝜎2}+𝑒^{−[𝑥𝑠𝑖+𝑣𝑇𝑖𝜉]2/2𝜎2}+78⋅1−𝑒^{−[𝑥𝑠𝑖+𝑣𝑇𝑖𝜉]2/((1/8)𝜎2)}. (27)

Signal reconstruction based on the solution of the problem in (28) works well, but the technique can be considerably enhanced by incorporating a reweighting strategy, and the reweighted unconstrained problem can be formulated as follows [21]: minimize_𝜉^𝑁_𝑖=1𝑤_𝑖⋅𝑓_𝜎𝑥_𝑠𝑖+𝑉^𝑇_𝑖𝜉, (28) where 𝑤_𝑖 are positive scalars that are from a weight vector 𝑤=[𝑤₁𝑤₂⋯𝑤_𝑁]. Starting with an initial 𝑤⁰=𝑒_𝑁, where 𝑒_𝑁 is the all-one vector of dimension 𝑁, in the (𝑘+1)th iteration the weight vector is updated to 𝑤^(𝑘+1) with its 𝑖th component given by 𝑤_𝑖^(𝑘+1)=1𝑥_𝑖^(𝑘),+𝜏 (29) where 𝑥_𝑖^(𝑘) denotes the 𝑖th component of vector 𝑥^(𝑘) obtained in the 𝑘th iteration as 𝑥^(𝑘)=𝑥_𝑠+𝑉_𝑛𝜉^(𝑘) and 𝜏 is a small positive scalar which is used to prevent numerical instability when |𝑥_𝑖^(𝑘)| approaches zero. As long as 𝜎>0, the objective function in (28) remains differentiable and its gradient can be obtained as 𝑔_𝑖(𝑥)=𝑤_𝑖18⋅4𝑥_𝑠𝑖+𝑣^𝑇_𝑖𝜉𝜎²+78⋅16𝑥_𝑠𝑖+𝑣^𝑇_𝑖𝜉𝜎²𝑒^{−8[𝑥𝑠𝑖+𝑣𝑇𝑖𝜉]2/𝜎2}. (30) For a fixed value of 𝜎, the problem in (26) is solved by using a quasi-Newton algorithm where an approximation of the inverse of the Hessian is obtained by using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update formula. It can be shown that function (28) remains convex in the region where the largest magnitude of the components of 𝑥=𝑥_𝑙2+𝑉_𝑛𝜉 is less than 𝜎, where vector 𝑥_𝑠 is chosen to be the least squares solution 𝑥_𝑙2 of 𝑦=Ψ𝑥; namely, 𝑥_𝑠=𝑥_𝑙2=Ψ^𝑇(ΨΨ^𝑇)⁻¹𝑦. Based on this, a reasonable initial value of 𝜎 can be chosen as 𝜎₀=max|𝑥_𝑙2|+𝜏, where 𝜏 is a small positive scalar. As the algorithm starts at the origin 𝜉⁽⁰⁾=0, the above choice of 𝜎₀ ensures that the optimization starts in a convex region. This greatly facilitates the convergence of the proposed algorithm.

Algorithm 3. The reweighted approximate smoothed 0 norm (RASL0 algorithm) is as follows.(1)Initialization: input Ψ, 𝑥_𝑙2, 𝜎_𝑇, 𝑟, 𝜏, and 𝜀, and set 𝜉⁽⁰⁾=0, 𝑤⁽⁰⁾=𝑒_𝑁, 𝜎=max|𝑥_𝑙2|+𝜏, and 𝑘=0.(2)Perform the 𝑄𝑅 decomposition of Ψ^𝑇=𝑄𝑅 and construct 𝑉_𝑛 using the last columns of 𝑄.(3)With 𝑤=𝑤^(𝑘) and 𝜉⁽⁰⁾ as an initial point, apply the BFGS algorithm to solve the problem in (28), where reweighting with parameter 𝜀 is applied using (29) in each iteration; denote the solution by 𝜉^(𝑘).(4)Compute 𝑥^(𝑘)=𝑥_𝑙2+𝑉_𝑛𝜉^(𝑘) and update weight vector to 𝑤^(𝑘+1) with (29).(5)If 𝜎≤𝜎_𝑇, stop and output 𝑥^(𝑘) as the solution; otherwise set 𝜉⁽⁰⁾=𝜉^(𝑘), 𝜎=𝑟⋅𝜎, 𝑘=𝑘+1, and repeat from step (3).

4. Simulation Result

This section describes some experiments testifying to the performances of the proposed algorithm. All experiments are performed under Windows 7 and MATLAB V8.0 (R2012a) running on a Lenovo workstation with an Intel (R) Core (TM), CPU i7 at 2.40 GHz, and 6 GB of memory.

Simulation 1: Algorithm Performances Comparison of One-Dimensional Signal. In the experiment, the signal length and number of measurements were set to 𝑁=256 and 𝑀=100, respectively. A total of 16 sparse signals with sparsity 𝐾=4𝑞−3, 𝑞=1,2,…,16, were used. Then a 𝐾-sparse signal 𝑥 was constructed as follows.(1)Set 𝑥 to zero vector of length 𝑁.(2)Generate a vector 𝑢 of length 𝐾 assuming that each component 𝑢_𝑖 is a random value drawn from a normal distribution 𝑁(0,1).(3)Randomly select 𝐾 indices from the set 1,2,…,𝑁, and set 𝑥_𝑖1=𝑢₁, and 𝑥_𝑖2=𝑢₂,…,𝑥_𝑖𝐾=𝑢_𝐾. The measurement matrix was assumed to be of size 𝑀×𝑁 and was generated by drawing its elements from 𝑁(0,1), followed by a normalization step to ensure that the 2 norm of each column is unity.

The performance of iteratively reweighted (IR) algorithm [22, 23] with 𝑝=0.1, the SL0 algorithm, ASL0 algorithm and RASL0 algorithm with 𝜎_𝑇=10⁻⁴, 𝑟=1/3, 𝜏=0.01, and 𝜖=0.08 was measured in terms of the number of perfect reconstructions over 100 runs. The results obtained were plotted in Figure 2. It can be observed that the RASL0 algorithm outperforms the IR algorithm. On comparing the SL0 algorithm and the ASL0 algorithm, we note that the two algorithms are comparable for 𝐾 smaller than 45 but the RASL0 algorithm performs better for 𝐾 larger than 45.

Figure 2 

Number of perfect reconstructions by each algorithm over 100 runs with 𝑁=256 and 𝑀=100.

The mathematical complexity of the SL0 algorithm, IR algorithm, ASL0 algorithm, and RASL0 algorithm was measured in terms of average CPU time over 100 runs for typical instances with 𝑀=𝑁/2 and 𝐾=round(𝑀/2.5) where 𝑁 varies in the range between 128 and 512. In Figure 3, it is noted that the RASL0 algorithm and the SL0 algorithms were more efficient than the IR algorithm, and the complexity of the RASL0 algorithm was slightly higher than that of the SL0 algorithm. The moderate increase in the mathematical complexity of the RASL0 algorithm was primarily due to the fact that the objective function in (26) needed to be modified in each iteration using (29). Typically the RASL0 algorithm converges in a small number of iterations. Figure 4 showed how the objective function in (26) converges in 24 iterations, where the parameters were set to 𝑁=256, 𝑀=100, 𝐾=40, and 𝜎=0.0212.

Figure 3 

Average CPU time required by each algorithm over 100 runs with 𝑀=𝑁/2 and 𝐾=round(𝑀/2.5).

Figure 4 

Function 𝐹𝜎(𝜉) for 𝑁=256, 𝑀=100, 𝐾=40, and 𝜎=0.0212.

Simulation 2: Algorithm Performances Comparison of Two-Dimensional Signal. As to examine the algorithms’ performances for different images, the reconstruction performance of each method is evaluated in terms of two indicators: the reconstruction relative error (RE) and peak signal-to-noise ratio (PSNR). The relative error of reconstructed image is defined as follows: RE(𝑥)=‖𝑥−𝑥‖₂‖𝑥‖₂, (31) where 𝑥 and 𝑥 are the original and the reconstructed image, respectively. Apparently, the lower the value of relative error is, the better the reconstructed performance will be. The PSNR of reconstructed images is defined by PSNR(𝑥)=10log₁₀𝑚×𝑛×255²‖𝑥−𝑥‖₂, (32) where 𝑚, 𝑛 are the size of the image.

In experiment, the DCT basis is selected as the sparse representation dictionary; compression ratio is 𝑀/𝑁=0.25. Figure 5 shows that the quality of the reconstruction with RASL0 algorithm has been improved and details of images are better reconstructed.

(a) Original image

(b) SL0

(c) ASL0

(d) RASL0

(e) Original image

(f) SL0

(g) ASL0

(h) RASL0

Figure 5 

The quality of the reconstruction with different algorithms in different images. Original image.

Table 1 shows three different 256×256 images, with the same compression ratio 𝑀/𝑁=0.5. From the comparison of the PSNR, the RE, the signal-to-noise ratio (SNR), and the matching degree (MD) among RASL0 algorithm, SL0 algorithm, and ASL0 algorithm, it is clear that the relative errors with RASL0 algorithm are 0.02% smaller than the SL0 algorithm, the SNR ratio is 3 dB higher, the PSNR is improved by 2 db, and the matching degree is also improved.

Table 1 

Reconstruction quality comparison with different images and different approximation smoothed 0 algorithms.

Figure 6 is the reconstruction effect comparison of image (Lena) with different algorithms. It shows that RASL0 outperforms OMP, GP, GPSR, and NSL0 algorithms. Table 2 is the quality comparison of different algorithms, indicating that RASL0 algorithm is better in all aspects.

Table 2 

Reconstruction quality comparison with four other algorithms.

(a) Original image

(b) OMP

(c) GP

(d) SP

(e) GPSR

(f) RASL

Figure 6 

Reconstruction quality comparison with four other algorithms.

5. Conclusion and Future Work

In this paper, we use approximate hyperbolic tangent function to estimate 0 norm and design a RSNL0 algorithm based on minimizing an approximate 0 norm of the signal in the null space of the measurement matrix, where a reweighting technique is used to force the solution’s sparsity and a quasi-Newton algorithm is used to accelerate the optimization. Simulation results are presented which demonstrate that the proposed algorithm yields improved signal reconstruction performance and requires a reduced amount of computation relative to iteratively reweighted algorithms based on the 𝑝 norm, with 𝑝<1. When compared with a known algorithm based on a smoothed 0 norm, improved signal reconstruction is achieved although the amount of computation is increased somewhat. However, how to choose the parameter decreasing sequence so as to eliminate the disturbance of the minimal value to the solution of the minimum value would be our future work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Basic Research Program of China (973 Program) under Grant no. 2013CB329404, the Major Research Project of the National Natural Science Foundation of China under Grant no. 91230101, the National Natural Science Foundation of China under Grant no. 61075006, and the Key Project of the National Natural Science Foundation of China under Grant no. 11131006.